Anahelixis

Geometric Realization of Second Integral Cohomology Classes The second integral cohomology classes in \( H^{2}(M ; Z) \) are in natural \( 1-1 \) correspondance with the isomorphism classes of circle-bundles over \( M \), where $M$ is a manifold. In particular, every second integral cohomology class \( \hat{F} \in H^{2}(M ; Z) \) has a geometric realization which consists of a space \( P=P_{F} \) together with an action of the circle \( S^{1} \) on \(P \) and a natural projection: \[\begin{array} \\ P \\ \downarrow \pi \\ M \end{array}\] so that the \( S^{1} \) action on \( P \) is free, preserves \( \pi, \) and \( \pi \) induces an isomorphism of \( P / S^{1} \) with \( M .\) For instance if \( M \) is a Riemann surface, consider that $P$ is the set of its unit tangent vectors to $M$, and let \( \pi \) be the projection map which assigns to each tangent vector the corresponding point of tangency. The circle \( S^{1} \) then acts freely by rotating any vector counter-clockwise in its...

Spectral Decomposition with Respect to a Circle Consider that \( T \) is an endomorphism of a finite-dimensional vector space \( E \), and let \( S \) be a circle in the complex plane which does not pass through any eigenvalue of \( T \). Then we consider the operator: \[\mathrm{Q}=\frac{1}{2 \pi i} \int_{S}(z-T)^{-1} d z\] $\mathrm Q$ is a projection operator in \( E \) which commutes with \( T . \) The decomposition \[E=E_{+} \oplus E_{-}, \quad E_{+}=Q E, \quad E_{-}=(1-Q) E\] is therefore invariant under \( T \), so that we can write: \[T=T_{+} \oplus T_{-}\] Then \( T_{+} \) has all eigenvalues inside \( S \) while \( T_{-} \) has all eigenvalues outside \( S \). This is just the spectral decomposition of \( T \) corresponding to the two components of the complement of the circle \( S \).

Complex Logarithm and Winding For any path \( \mathcal{C} \)  and any point \( a \) not on \( \mathrm {C}, \) the complex logarithm enables us to define the following function: \[\begin{array}{l} \quad N(\mathcal{C} ; a)=\frac{1}{2 \pi i} \int_{\mathrm{C}} \frac{d z}{z-a}=\frac{1}{2 \pi i} \int_{z} d \log (z-a)=\left.\frac{1}{2 \pi} \arg (z-a)\right|_{\mathrm C} \end{array}\] This is called the winding number of the path \( \mathrm{C} \) with respect to \( a \).  Since the indeterminacy of the argument is in multiples of \( 2 \pi \), if \( \mathrm{C} \) is a closed path, then the values of \( N(\mathrm{C} ; a) \) are integers. The expression \( N(\mathrm{C} ; a) \) is simultaneously a function of both \( \mathrm{C} \) and \( a \). Evidently, \( N \) depends continuously on \( a \) and $\mathrm C$ unless \( a \) reaches \( \mathrm{C} . \) This means that if a family of paths \( \mathcal{C}_{t} \), where none of them is passing through \( a \), were parametrized continuously b...